# $L$-function of induced Galois representation

Suppose $L \subset K$ are number fields and let's choose and fix an algebraic closure $\overline{L}$ of $L$ such that $L \subset K \subset \overline{L}$, hence $H:=\text{Gal}(\overline{L}/K)$ is a closed subgroup of $G:=\text{Gal}(\overline{L}/L)$ of finite index $[K, L]$. Suppose now we have an $\ell$-adic representation $V$ of $\text{Gal}(\overline{L}/K)$, then there will exist an induced representation $\text{Ind}_H^G\,V$ of $\text{Gal}(\overline{L}/L)$. If necessary we could assume $V$ has a geometric origin, e.g. it is the etale cohomology of a smooth variety defined over $K$.

For a finite prime $\mathfrak{P}$ of $K$, the local $L$-factor is defined as $$L_{\mathfrak{P}}(V,s):=\text{det}(1-\text{Fr}_{\mathfrak{P}}(N(\mathfrak{P}))^{-s}\big|V^{I_{\mathfrak{P}}})$$ where $I_{\mathfrak{P}}$ is the inertia group at $\mathfrak{P}$. Similarly we could define the local $L$-factor for a prime $\mathfrak{p}$ of $L$. Question If $V$ is an Artin representation, we have $$L_{\mathfrak{p}}(\text{Ind}_H^G\,V,s)=\prod_{\mathfrak{P}|\mathfrak{p}}L_{\mathfrak{P}}(V,s)$$ I guess it is still true for $\ell$-adic representations, but I don't know how to prove. The method to prove this formula might still work, since the Galois groups $H$ and $G$ are locally compact, but I am not sure as I know basically nothing about representation theories of locally compact groups. It would be great if someone could give a complete proof.

• Are you assuming $\mathfrak{P}$ has residue characteristic $p \ne \ell$? I'm pretty sure this is not true if $p = \ell$. – David Loeffler Mar 21 '18 at 8:31
• @DavidLoeffler Yes, let's assume $p \neq \ell$! – Wenzhe Mar 21 '18 at 10:23
• Key points: (i) if $f:X\to Y$ is a finite scheme map (e.g., ${\rm{Spec}}(O_K)\to {\rm{Spec}}(O_L)$) then for a geometric point $y\in Y$ and abelian sheaf $F$ on $X_{et}$ we have $f_{\ast}(F)_y=\prod_{x\in f^{-1}(y)} F_x$, (ii) if $j:\eta\to X$ is the generic point of a connected Dedekind $X$ and $G$ is an abelian sheaf on $\eta_{et}$ then $j_{\ast}(G)_x=G(\overline{\eta})^{I_x}$ for any geometric closed point $x$ and an inertia group $I_x\subset {\rm{Gal}}(\overline{\eta}/\eta)$, (iii) Ind = finite pushforward (adjoint to "Res = pullback") via "Galois module = etale sheaf". And $p=\ell$ is OK! – nfdc23 Mar 21 '18 at 13:44
• @nfdc23 Thank you, I will have a try. – Wenzhe Mar 21 '18 at 14:14
• Hi Wenzhe, did you find a reference for this fact? – Med Jun 27 at 2:09